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### Lesson 3-7

Higher Order Deriviatives

Objectives

- Find second and higher order derivatives using all previously learned rules for differentiation

Vocabulary

- Higher order Derivative – taking the derivative of a function a second or more times

Example 1

Find second derivatives of the following:

- y = 5x³ + 4x² + 6x + 3
- y = x² + 1

y’(x) = 15x² + 8x + 6

y’’(x) = 30x + 8

y’(x) = ½ (x² + 1)-½ (2x) = x / (x² + 1)½

(1)(x² + 1)½ - x ½ (x² + 1) -½ (2x)

y’’(x) = --------------------------------------------

(x² + 1)

Example 2

Find the fourth derivatives of f(x) = (1/90)x10 + (1/60)x5

Find a formula for f n(x) where f(x) = x-2

f’(x) = (10/90)x9 + (5/60)x4

f’’(x) = (90/90)x8 + (20/60)x3

f’’’(x) = (720/90)x7 + (60/60)x2

F’’’’(x) = (5040/90)x6 + (120/60)x = 56x6 + 2x

f’(x) = (-2)x-3

f’’(x) = (-2)(-3)x-4

f’’’(x) = (-2)(-3)(-4)x-5

f’’’’(x) = (-2)(-3)(-4)(-5)x-6

fn(x) = (-1)n(n+1)! x-(2-n)

Example 3

Find the third derivatives of the following:

y = (3x³ - 4x² + 7x - 9)

f(x) = e2x+7

y’(x) = 9x² - 8x + 7

y’’(x) = 18x - 8

y’’’(x) = 18

f’(x) = 2e2x+7

f’’(x) = 4e2x+7

f’’’(x) = 8e2x+7

Chain Rule Revisited

FunctionDerivativeFunctionDerivative

y = xn y’ = n xn-1 y = un y’ = n un-1 u’

y = ex y’ = ex y = eu y’ = u’ eu

y = sin x y’ = cos x y = sin u y’ = u’ cos u

y = cos x y’ = -sin x y = cos u y’ = -u’ sin u

y = tan x y’ = sec² x y = tan u y’ = u’ sec² u

y = cot x y’ = -csc² x y = cot u y’ = u’ csc² u

where u is a function of x (other than just u=x) and u’ is its derivative

Derivatives FAQ

- When do I stop using the chain rule?Answer: when you get something that you can take the derivative of without having to invoke the chain rule an additional time (like a polynomial function).
- Does the argument in a trig function ever get changed?No. The item (argument) inside the trig function never changes while taking the derivative.
- Does the exponent always get reduced by one when we take the derivative?Only if the exponent is a constant! If it is a function of x, then it will remain unreduced and you have to use another rule instead of simple power rule.
- When do I use the product and quotient rules?Anytime you have a function that has pieces that are functions of x in the forms of a product or quotient.

Summary & Homework

- Summary:
- Derivative of Derivatives
- Use all known rules to find higher order derivatives

- Homework:
- pg 240 - 242: 5, 9, 17, 18, 25, 29, 49, 57

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